which means that the Universe is empty.
Finally, the investigation of GWs in PCG is also inconsistent with observations.
The linearized version of eq. (246) in the Lorenz gauge ∂ρˆhµρ = 0 results in the wave equation [194]
2ˆhµν =− 1
2αgTµν, (259)
where ˆhµν =hµν− 14ηµνh . The vacuum solutions are
ˆhµν =Aµνe±ikρxρ +Bµνnρxρe±ikρxρ, (260) whereAµν andBµν are polarization tensors constrained by the harmonic gauge. ˆhµν is given by a sum of a plane wave (which is also a solution tohµν = 0 as in GR) and a wave that grows linearly in time and with distance representing Ostrogradsky’s instability (cf. Sec. 6.7). This would lead to
The inhomogeneous solution of eq. (259) reads [194]
hˆµν = 1 16παg
Z
d4x0Θ(t−t0)Θ [(t−t0)− |x−x0|]Tµν(x0). (261) We see that the amplitude does not decrease with the distance, which differs sig-nificantly from the 1/r dependence of GWs in GR. Besides that, the GWs are supported on the whole past light cone. This is obviously in contradiction with observations since it violates the constraints on the speed of GWs which we derived in Sec. 6.9. See also [195] for an interesting method to derive the GW solutions.
All these problems together imply that PCG cannot be the final answer. There-fore, in the next section we present an extension of PCG, which solves the problem with the traceless matter and leads to a non-empty cosmology, but on the other hand introduces a (tachyonic) ghost particle.
7.3 Extended Conformal Gravity 71
to gravity. As we discussed in Sec. 3.2, a nonminimal matter-curvature coupling actually implies that Einstein’s equivalence principle is violated, but after fixing the Weyl gauge it becomes apparent that Einstein’s equivalence principle still is valid.
Additionally, we use a spinor fieldψ(x) coupled toS(x) as a representative of the fermionic sector. We do not include gauge bosons or a Higgs sector here since it is not relevant for our analysis of GWs in (P2) and (P3). Hence, this action represents a toy model, but it can be easily extended to the standard model of particle physics, since before the gauge symmetry SU(3)×SU(2)×U(1) is spontaneously broken, i.e. before masses are generated via the Higgs mechanism, the standard model of particle physics is locally Weyl invariant. Therefore, we need to find a way to couple a locally Weyl invariant Higgs sector to the other standard model fields. We discuss this briefly in Sec. II of (P2). For a more detailed discussion of this issue, see e.g.
[196, 197, 198, 199].
The most general local matter action IM for the scalar field S and the spin-1/2
fermion field ψ is thus given by [180]
IM =− Z
d4x√
−g
−S,µS,µ
2 + S2R 12
+λS4+iψγ¯ µ(x) [∂µ+ Γµ(x)]ψ−ξSψψ¯
, (262) whereξandλare dimensionless coupling constants,γµ(x) are the vierbein-dependent Dirac-gamma matrices, ¯ψ = ψ†γ0 and Γµ(x) is the fermion spin connection64. To be invariant under LWTs the fields have to transform as shown in Sec. 7.1. We observe that the scalar field couples to the Ricci scalar in a similar way as in the scalar-tensor theory, which we presented in Sec. 6.2.1. Thus, the action in eq. (262) is given in the Jordan frame.
Note that only the combination of the two terms in the parentheses is Weyl invariant. Hence, we introduced the parameter , which can assume values of −1 or +1, in eq. (262). In the first case, the theory corresponds to CG, while in the second it corresponds to a Weyl invariant version of Einstein-Weyl gravity, which after fixing the Weyl gauge resembles Einstein-Weyl gravity [200], as will become clear later65.
For R < 0 and λ > 0 the potential V(S) = S2R/12 +λS4 can lead to a spontaneous breaking of Weyl symmetry, although in our simple toy model we do not need to break the Weyl symmetry since we can just fix it by a gauge condition.
Nevertheless, as discussed above in a more realistic version of the matter action, including also additional scalar fields, this could become relevant since we can only fix one dof via the Weyl gauge. Hence, this is an interesting topic for further studies.
Variation of eq. (262) with respect to S and ψ leads to the field equations
−S,µ;µ− 1 6SR
−4λS3+ξψψ¯ = 0, (263) iγµ(x) [∂µ+ Γµ(x)]ψ−ξSψ= 0. (264)
64We have used a simplified notation. The kinetic part of the fermionic action is given by iψγ¯ µ(x) [∂µ+ Γµ(x)]ψ−iψ¯h←−
∂µ+ Γµ(x)i
γµ(x)ψin order to be Hermitian.
65Note that this case is called massive conformal gravity (MCG) in (P2). But it was realized that there is a very similar, but still different theory with the name massive conformal gravity. To avoid confusion we will not use this name in this thesis.
Variation of the action given in eq. (245) in connection with eq. (262) with respect to gµν leads formally to the Bach equations as in eq. (246), but with a modified matter energy-momentum tensor. Using eq. (264) the matter energy-momentum tensor can be written as
TµνM =Tµνf +
−2S,µS,ν
3 +gµνS,αS,α
6 + SS,µ;ν
3 − gµνSS,α;α
3 + 1
6S2Gµν
−gµνλS4, (265) where
Tµνf ≡ 1 2
iψγ¯ µ(x) [∂ν + Γν(x)]ψ+ (µ↔ν)
(266) is the energy-momentum tensor of the fermion. Note that in eq. (265) the Einstein tensor appears. This is an important difference to PCG since the complete trace of TµνM has to vanish and hence the fermionic part, representing the standard model of particle physics, does not have to be traceless. The consequences of this will be discussed below.
Because of the local Weyl invariance, it is always possible to choose a frame in which the scalar field is constant
S(x)→S0(x) = Ω−1(x)S(x) = S0 = const., (267) with Ω(x) = S(x)/S0. We call this theunitary gauge. This points out that the scalar field S(x) is just an auxiliary field and we do not need to worry about its stability properties [196,201]. Nevertheless, in Appendix C of (P2) we briefly discuss ghosts and tachyons for a scalar field. For a detailed discussion on the ghost issue, see also [123]. In order to connect this theory to GR, i.e. to see similarities and differences, we choose the scalar field such that the coefficient in front of the Einstein tensor becomes that of the EFE (multiplied by ):
8πG≡ 6
S02, (268)
Λ≡6λS02, (269)
where G denotes Newton’s constant and Λ is the cosmological constant. For a constant scalar field all terms with derivatives on S vanish and thus the matter energy-momentum tensor reduces to
TµνM =Tµνf +
8πGGµν− Λ
8πGgµν. (270)
We observe that we have transformed the theory into the Einstein frame represen-tation, since we have eliminated the dependence on the scalar field. In the unitary gauge, fermions have a constant mass given by mf = ξS0. Since it is known from experiments that fermions have positive masses, we chooseξS0 >0. In consequence, eqs. (263) and (264) read
−R+ 4Λ
8πG +mfψψ¯ = 0, (271)
Tf −mfψψ¯ = 0, (272)
7.3 Extended Conformal Gravity 73
where Tf denotes the trace of the fermion energy-momentum tensor. Combining these two equations we find
R+ 4Λ = 8πGTf. (273)
We note that it is convenient to introduce a ”graviton mass”mg via m2g ≡ 1
32πGαg. (274)
Besides having the dimensions of a mass, at this point it is not obvious thatmg does indeed play the role of a mass for the graviton. This will become clear in Sec. III of (P2). Using eqs. (270) and (274) in eq. (246) we obtain [202,203]
−Gµν +gµνΛ + 1
m2gWµν = 8πGTµνf . (275) The limit which reproduces the EFE is given by mg → ∞ (αg → 0) for = +1.
This is equivalent to IW = 0 in eq. (245). For a detailed discussion of the limits of CGMs, see Table I in (P2) and the text below it. Observe that the fermion energy-momentum tensor is covariantly conserved,
Tfµρ;ρ= 0, (276)
due to the Bianchi identities for the Bach and Einstein tensors. This means that test particles move on geodesics and nongravitational physics is locally Lorentz invariant.
Thus, Einstein’s equivalence principle becomes manifest in the unitary gauge. For a more detailed derivation of the field equations, see Sec. II of (P2).
Looking at eq. (275) it is obvious that this extended model of CG also improves the field equations for FLRW spacetimes. The Bach tensor vanishes in conformally flat spacetimes and thus, for= +1 eq. (275) agrees with the EFE for isotropic and homogeneous FLRW models. If we consider cosmological perturbation theory, the situation is different. The perturbation of the Bach tensorδWµν does not vanish and hence there appear modifications to the linearized Friedmann equations [204, 205].
For=−1 the Einstein tensor has the wrong sign and thus gravity is repulsive66. In this case the composition of the Universe has to be very different from the ΛCDM model. Nevertheless, it is claimed that the Hubble diagram can be explained for the following density parameters of the current Universe [8, 206]: matter density parameter ΩM = O(10−60), curvature density parameter Ωk = 0.63 and dark en-ergy density parameter ΩΛ = 0.37. On this other hand, an analysis of gamma ray bursts and quasars exhibits that ΛCDM is favored on high redshifts by the data [207]. Besides that, cosmological problems, like the singularity, horizon, flatness and cosmological constant problems, can be solved [8, 180, 204, 208, 209, 210]. It turns out that the Universe must be open (negative curvature) and the decelera-tion parameter is always negative [208], which means it always expands accelerated.
However, to be consistent with the expansion rate observed today, the expansion during primordial nucleosynthesis must have been much slower than in the standard FLRW model. Consequently, the deuterium burning lasted much longer, imply-ing that almost no deuterium is left, which is in contradiction with lower limits on
66Note that the gravitatioanl force on local scales is still attractive and hence there is no obvious contradiction with SS tests.
the deuterium abundance [211, 212]. On the other hand, conformal cosmology is singularity-free meaning that the Universe has a minimum size [208]. Moreover, there is no analysis of the cosmic microwave background yet. Only some early stud-ies on cosmological perturbation theory have been worked out [204, 205], but no compelling results are derived yet. On top of that, structure formation has not been investigated yet, and it is not clear if it is possible to explain the formation of larger structures, like galaxies and galaxy clusters, without dark matter. Nevertheless, these results are still under debate [207].
Concerning the gravitational potential, another problem immediately appears.
In PCG we have used the gravitational potential in eq. (255) to fit galaxy rotation curves. To derive this potential it was necessary to choose a specific Weyl gauge. If we choose the same gauge in the extended CGMs, we cannot set the scalar field to a constant as in the unitary gauge. The scalar field would be dynamical and masses of standard model particles would depend on spacetime coordinates; cf. eq. (264).
Thus, it is not clear whether these models still can fit galaxy rotation curves without dark matter. This is heavily debated in the literature [201,213, 214, 215, 216, 217, 218, 219].